Pushforward (differential)


In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of at a point , denoted , is, in some sense, the best linear approximation of near . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of at to the tangent space of at , . Hence it can be used to push tangent vectors on forward to tangent vectors on . The differential of a map is also called, by various authors, the derivative or total derivative of .

"If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N."
If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N.

Motivation edit

Let   be a smooth map from an open subset   of   to an open subset   of  . For any point   in  , the Jacobian of   at   (with respect to the standard coordinates) is the matrix representation of the total derivative of   at  , which is a linear map


between their tangent spaces. Note the tangent spaces   are isomorphic to   and  , respectively. The pushforward generalizes this construction to the case that   is a smooth function between any smooth manifolds   and  .

The differential of a smooth map edit

Let   be a smooth map of smooth manifolds. Given   the differential of   at   is a linear map


from the tangent space of   at   to the tangent space of   at   The image   of a tangent vector   under   is sometimes called the pushforward of   by   The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space).

If tangent vectors are defined as equivalence classes of the curves   for which   then the differential is given by


Here,   is a curve in   with   and   is tangent vector to the curve   at   In other words, the pushforward of the tangent vector to the curve   at   is the tangent vector to the curve   at  

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by


for an arbitrary function   and an arbitrary derivation   at point   (a derivation is defined as a linear map   that satisfies the Leibniz rule, see: definition of tangent space via derivations). By definition, the pushforward of   is in   and therefore itself is a derivation,  .

After choosing two charts around   and around     is locally determined by a smooth map   between open sets of   and  , and


in the Einstein summation notation, where the partial derivatives are evaluated at the point in   corresponding to   in the given chart.

Extending by linearity gives the following matrix


Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map   at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from   to  . In general, the differential need not be invertible. However, if   is a local diffeomorphism, then   is invertible, and the inverse gives the pullback of  

The differential is frequently expressed using a variety of other notations such as


It follows from the definition that the differential of a composite is the composite of the differentials (i.e., functorial behaviour). This is the chain rule for smooth maps.

Also, the differential of a local diffeomorphism is a linear isomorphism of tangent spaces.

The differential on the tangent bundle edit

The differential of a smooth map   induces, in an obvious manner, a bundle map (in fact a vector bundle homomorphism) from the tangent bundle of   to the tangent bundle of  , denoted by  , which fits into the following commutative diagram:


where   and   denote the bundle projections of the tangent bundles of   and   respectively.

  induces a bundle map from   to the pullback bundle φTN over   via


where   and   The latter map may in turn be viewed as a section of the vector bundle Hom(TM, φTN) over M. The bundle map   is also denoted by   and called the tangent map. In this way,   is a functor.

Pushforward of vector fields edit

Given a smooth map φ : MN and a vector field X on M, it is not usually possible to identify a pushforward of X by φ with some vector field Y on N. For example, if the map φ is not surjective, there is no natural way to define such a pushforward outside of the image of φ. Also, if φ is not injective there may be more than one choice of pushforward at a given point. Nevertheless, one can make this difficulty precise, using the notion of a vector field along a map.

A section of φTN over M is called a vector field along φ. For example, if M is a submanifold of N and φ is the inclusion, then a vector field along φ is just a section of the tangent bundle of N along M; in particular, a vector field on M defines such a section via the inclusion of TM inside TN. This idea generalizes to arbitrary smooth maps.

Suppose that X is a vector field on M, i.e., a section of TM. Then,   yields, in the above sense, the pushforward φX, which is a vector field along φ, i.e., a section of φTN over M.

Any vector field Y on N defines a pullback section φY of φTN with (φY)x = Yφ(x). A vector field X on M and a vector field Y on N are said to be φ-related if φX = φY as vector fields along φ. In other words, for all x in M, x(X) = Yφ(x).

In some situations, given a X vector field on M, there is a unique vector field Y on N which is φ-related to X. This is true in particular when φ is a diffeomorphism. In this case, the pushforward defines a vector field Y on N, given by


A more general situation arises when φ is surjective (for example the bundle projection of a fiber bundle). Then a vector field X on M is said to be projectable if for all y in N, x(Xx) is independent of the choice of x in φ−1({y}). This is precisely the condition that guarantees that a pushforward of X, as a vector field on N, is well defined.

Examples edit

Pushforward from multiplication on Lie groups edit

Given a Lie group  , we can use the multiplication map   to get left multiplication   and right multiplication   maps  . These maps can be used to construct left or right invariant vector fields on   from its tangent space at the origin   (which is its associated Lie algebra). For example, given   we get an associated vector field   on   defined by

for every  . This can be readily computed using the curves definition of pushforward maps. If we have a curve
we get
since   is constant with respect to  . This implies we can interpret the tangent spaces   as  .

Pushforward for some Lie groups edit

For example, if   is the Heisenberg group given by matrices

it has Lie algebra given by the set of matrices
since we can find a path   giving any real number in one of the upper matrix entries with   (i-th row and j-th column). Then, for
we have
which is equal to the original set of matrices. This is not always the case, for example, in the group
we have its Lie algebra as the set of matrices
hence for some matrix
we have
which is not the same set of matrices.

See also edit

References edit

  • Lee, John M. (2003). Introduction to Smooth Manifolds. Springer Graduate Texts in Mathematics. Vol. 218.
  • Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag. ISBN 3-540-42627-2. See section 1.6.
  • Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. London: Benjamin-Cummings. ISBN 0-8053-0102-X. See section 1.7 and 2.3.